Geometry and physics of wrinkling

Overview

Authors: E.Cerda & L.Mahadevan

Source: Physical Review Letters, Vol.90, 7, (2003)

Soft Matter key words: wrinkling, elastic sheet, tension, compression

Abstract

In this publication, authors set out to develop a general theory of wrinkling, using elementary geometry and the physics of bending and stretching. As a result they produce scaling laws for the wrinkle wavelength <math>\lambda</math> and amplitude A. They proceed to test these scaling laws for various wrinkling circumstances: the wrinkling of a polyethylene sheet, the wrinkling of an apple, the wrinkling of human skin and even the wrinkling of polymerized vesicles used for drug delivery.

Soft Matter Snippet

Fig.1 : E.Cerda & L.Mahadevan

Fig.2 : F.Brochard-Wyart & P.G.de Gennes

The derivation starts off by considering the stretching of a polyethylene sheet , clamped at the edges. Beyond a critical stretching strain the sheet wrinkles as depicted on figure 1. The functional for this process is:

<math>U = U_B + U_S -L</math>

Where <math>U_B</math> is the bending energy due to deformations on the y axis and is a function of the bending stiffness B. Accordingly, <math>U_S</math> is the stretching energy due to tension T(x) along the x direction. Variable L represents the condition of inextensibility that the sheet has to satisfy. Manipulating the equation leads the authors to an expression for U:

<math>U = B \kappa^2_n \Delta L + \pi^2 T \Delta /\kappa^2_n L</math>

Here <math>\Delta</math> is the imposed compressive transverse displacement and <math>\kappa_n</math> is the wave number. The wavelength and amplitude are obtained by minimizing <math>U</math>:

Where the scaling law for the wavelength arises by a substitution of the tension-to-length ratio by the stiffness K of the elastic foundation: <math>K \sim \frac{T}{L^2}</math>

And now for the fun part! The authors test their model for skin wrinkling, a rather unusual soft matter application! They start off with the observation that the wavelength of human wrinkles is larger than both the elastic substrate thickness <math>H_s</math> on which the skin rests, as well as the thickness of the skin itself t. That is: <math>\lambda >> H_s >> t</math>. The wrinkle stiffness is of the order of: